Question

An approach at a pretimed signalized intersection has a saturation flow rate of 2000 veh/h and is allocated 15 seconds of effective green in a 60-second signal cycle. If the flow at the approach is 300 vel/h, provide an analysis of the intersection assuming D/D/1 queuing.

 a. The time to queue clearance after the start of the effective green

 b. The Proportion of cycle with a queue

 c. The proportion of vehicles stopped

 d. The Maximum number of vehicles in the queue

 e. The total vehicle delay per cycle

 f. The average delay per vehicle

 g. The maximum delay of any vehicle

Answer 1

We are given saturation flow rate = 2000 veh/hour

We are given that arrival flow is 300 veh/hour, and departure flow once the signal turns green is at max 2000 veh/hour.

The signal is green for 15 seconds out of 60 seconds. So lets set up a queuing table

Arrival Flow is 300 veh/hr, so in 15 seconds we have 1.25 arrivals. Arrivals happen in green and red

Departure flow max = 2000 veh/hr, so in 15 seconds we have 8.33 departures, BUT DEPARTURES HAPPEN ONLY IN GREEN.

If the accumulated queue at start of green is less than 8.333, then entire queue will be cleared, else the difference above 8.33 will be queued up.

Time Range (secs) Red/Green Start Time Arrival Flow Rate(veh/hr) # of arrivals in 15 seconds Cumulative Arrivals Departure Flow (veh/hr) RED 0 0 0 2000 0 GREEN 0-15 15-30 30-45 45-60 60-75 75-90 90-105 105-120 120-135 135-150 150-165 165-180 300 300 300 300 300 300 300 300 300 300 0 15 30 45 60 75 90 105 120 135 150 165 1.25 2.5 3.75 5 1.25 2.5 3.75 5 RED 0 # of Queue length depatures Cumulative (veh) at end in 15 Departures of the time seconds interval 0.000 0.000 1.25 0.000 0.000 2.5 0.000 0.000 3.75 8.333 8.333 0 0.000 0.000 1.25 0.000 0.000 2.5 0.000 0.000 3.75 8.333 8.333 0 0.000 0.000 1.25 0.000 0.000 2.5 0.000 0.000 3.75 8.333 8.333 0 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.25 0 2000 GREEN 0 RED 0 1.25 2.5 3.75 5 300 0 2000 GREEN 300

We can plot it on a graph as follows

So in our case assume that we start with RED time

after 45 seconds we have 3.75 vehicles queued up, in 45-60 interval another 1.25 vehicles arrive. so total arrivals = 5(cumulative arrivals)

but in that green 8.33 vehicles are cleared, so the queue clears

So we can answer the questions

a) At start of effective green saturation flow = 2000 veh/hour and we have 5 vehicles which are queued up

so time to clear the queue of 5 vehicles = 5/ (2000/3600) = * 3600/2000 = 9 seconds

so time to clear queue after start of effective green = 9 seconds

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b) Proportion of a cycle with a queue. From the table in a cycle of 60 seconds there is a queue during 45 seconds.

so proportion of cycle with a queue = 45/60 = .75 = 75%

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c) Proportion of vehicles stopped.

Within a cycle we have 5vehicle arrivals. Out of these 3.75 are stopped

So proportion of vehicles stopped = 3.75/5 = 0.75

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d) Maximum number of vehicles in the Queue = 3.75 vehicles

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e) Total vehicle delay per cycle.

This is equal to # of stopped vehicles * (time for which they were stopped) = (1.25 * 15) + (1.25 * 15) + (1.25 * 15) = 56.25 vehicle seconds

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f) Average delay per vehicle = Total vehicle delay/number of vehicles = 56.25/3.75 = 15 seconds

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g) Maximum Delay of any vehicle. This will be 45 seconds

A vehicle which arrives at the start of the red time, will have to wait 45 seconds before green starts.

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